interval calculation

[vestal pebble]

1/A =? range

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[novel rain]

Note that f(x) = 1/x is only monotone separately for x > 0 and for x < 0 and is undefined at x = 0.
So, try separating your interval like [-2, 0)⋃{0}⋃(0, 7], then you can see how f(x) = 1/x acts on it.

[junior lantern]

The concept of "1 divided by an interval" is nonsensical.

[novel rain]

I think it's an informal way to write f(A), where f(x) = 1/x.

So, like, an image of f(x) when applied to that set.

[vestal pebble]

at x= 0 its undefined then too we hv to include 0?y

[vestal pebble]

ok

[novel rain]

Well, 0 doesn't go anywhere, of course.

So, just see what happens to the other two pieces.

[vestal pebble]

so are [-2, 0)⋃{0}⋃(0, 7] and [-2, 0)⋃(0, 7] the same?

no

[novel rain]

Well, no, but the point x = 0 doesn't have an image when f(x) = 1/x is applied to it, since the function is undefined at that point.

So, you just need to see what the function does to [-2, 0) and (0, 7].

If you're having the trouble with the part of the interval near zero, here's an approach (for the left interval as an example).
Take [-2, a), where a < 0. Apply f(x) = 1/x to it. And then find what happens when a -> 0-.

Same approach for the other interval, but the limit will be from the right there.

[junior lantern]

Actually, if the function 1/x isn't defined over the whole interval, does the image of the interval even exist, properly speaking? Like, from a set-theoretic perspective 1/x isn't even a function on any set that includes 0.

[novel rain]

Hm...

Good question. Not sure.

I'd just say that if x = x0 isn't in the domain of f(x), then f({x0}) = ∅.

[junior lantern]

Except that's not correct either, because domain and range are explicitly part of the definition of a function in set theory.

[vestal pebble]

my teacher told to do it like this, n idk how in the first interval 0 is present, then too its called negative interval, he said if u dont understand just rem the pattern for when a set is like this -->[+,-] one -ve n 1 +ve no. that -inf and inf will def come in the interval when u gotta solve it for 1/A from A

[vestal pebble]

lemme read this slowly again i need time

[novel rain]

Not sure if I like that subdivision. I think [-2, 0)⋃{0}⋃(0, 7] makes more sense, at least if you follow the convention I wrote above.

[vestal pebble]

can i get an example when its not defined over the whole interval, i dont understand the image thing, how to draw ''1/x isn't even a function on any set that includes 0.'' how does 0 affect the whole set

[junior lantern]

...this is the exact example. The interval includes 0, and the function f(x) = 1/x isn't defined at x = 0.

[vestal pebble]

ive tried solving it for a=-4 but i wrote it like this---(-1/4,-1/2] and idk how to do limits part

[vestal pebble]

uhh, okay getting it

[vestal pebble]

i feel so lost idk what to do

[novel rain]

Well, if X = [-2, a), where a < 0, then f(X) = (1/a, -1/2], right?

[vestal pebble]

im sorry i dont get it,my brain thinks a<0 means a negative number so i took a=-4 just as an example

-4 is less than -2 so i cant write the set like this

[junior lantern]

...-2 < a < 0.

[vestal pebble]

if i take a as -1?

its not working cant write (-1,-1/2]

[-1/2,-1)?

[novel rain]

f(x) = 1/x is a decreasing function for x < 0.

So, if you have X = [a, b], where a, b < 0, and f(x) = 1/x, then f(X) = [1/b, 1/a].

[novel rain]

So, you do this, then look at the limit as a -> 0-.

[vestal pebble]

what to do after this?

im getting big negative numbers

no 0

how to reach -inf

[vestal pebble]

i did something pls check

[vestal pebble]

for finding the range

[vestal pebble]

+close

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[novel rain]

What are you doing?

[vestal pebble]

trying to find range , earilier i wasnt able to understand how the negative interval is approaching -inf i was sleepy n im again, but i do now, ig? if its correct

[novel rain]

The rows should start with 0 < A ≤ 7 and -2 ≤ A < 0. And you messed up the second part in the answer a bit.

But the idea is correct.

[vestal pebble]

okay, got it

thanks a mil

[novel rain]

You're welcome!

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@vestal pebble

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